Finite Group Databasebeta

← #37544889

Subgroups — #37544889

Order 23168.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041813C211yesmaximal, characteristic
2#26041678C212yesmaximal, complemented
3#26041636C212yesmaximal, complemented
4#26041614C211yesmaximal, characteristic, complemented
5#26041609C211yesmaximal, characteristic, complemented
6#26041597C211yesmaximal, characteristic
7#26041592C211yesmaximal, characteristic
8#26041583C212yesmaximal, complemented
9#26041576C212yesmaximal, complemented
10#26041332C211yesmaximal, characteristic, complemented
11C2×#125×C181C211yesmaximal, characteristic, complemented
12#19153810—21no—
13#19153808C2211yescharacteristic
14#19153804C2211yescharacteristic
15#19153799—21no—
16#19153795C2211yescharacteristic
17#19153790C2211yescharacteristic
18C4×Dic362C2211yescharacteristic
19C4×D724C2211yescharacteristic
20C4×C22⋊D181—21no—
21#19153773—21no—
22#19153749C2212yes—
23#19153749C2212yes—
24#19153747C2212yes—
25#19153747C2212yes—
26C4⋊#13089678C2212yes—
27#19153739C2212yes—
28#19153739C2212yes—
29#19153736C2212yes—
30#19153736C2212yes—
31C22⋊#13089678C2211yescharacteristic
32C22⋊#13089678C2211yescharacteristic
33C4×#13089678C2212yes—
34#125×C181—21no—
35#125×C181—21no—
36#125×C181C2212yes—
37#125×C181C2212yes—
38C2×C4⋊C4×C181—21no—
39C2×C4⋊C4×C181—21no—
40C2×C4⋊C4×C181C2211yescharacteristic
41C2×C4⋊C4×C181C2211yescharacteristic
42C2×C42×C181C2211yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181—21no—
47C2×#13089678C2312yes—
48C2×#13089678C2312yes—
49C4⋊Dic181C2311yescharacteristic
50C4⋊Dic181C2311yescharacteristic
51#15668428—21no—
52#15668428—21no—
53#15668428—21no—
54#15668428—21no—
55C22⋊Dic181—21no—
56#15668426—21no—
57#15668426—21no—
58#15668426—21no—
59#15668426—21no—
60C4×Dic181—21no—
61C22×C4×C181—21no—
62C22×C4×C181—21no—
63C22×C4×C181D411yescharacteristic
64C22×C4×C181D411yescharacteristic
65C22×C4×C181C2311yescharacteristic
66C4⋊C4×C181—22no—
67C4⋊C4×C181—22no—
68C4⋊C4×C181—22no—
69C4⋊C4×C181—22no—
70C4⋊C4×C181—21no—
71C4⋊C4×C181—21no—
72C4⋊C4×C181C2312yes—
73C4⋊C4×C181C2312yes—
74C42×C181D411yescharacteristic
75C42×C181D411yescharacteristic
76C42×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78C22×D181—21no—
79Dic362—22no—
80D724—22no—
81C22⋊D181—42no—
82C2×Dic181—21no—
83C2×Dic181—21no—
84C2×Dic181—21no—
85C4×D181—42no—
86#13089678—22no—
87#13089678—22no—
88C23×C181C2×D411yescharacteristic
89C2×C4×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—22no—
92C2×C4×C181—22no—
93C2×C4×C181—41no—
94C2×C4×C181—42no—
95C2×C4×C181—42no—
96C2×C4×C181C2×D411yescharacteristic
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2×D411yescharacteristic
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2×D411yescharacteristic
101C2×C4×C181C2411yescharacteristic
102C22×C181—21no—
103C22×C181—21no—
104C22×C181#13512yes—
105C22×C181C22×D411yescharacteristic
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—41no—
109C4×C181—21no—
110C4×C181—21no—
111C4×C181—21no—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—42no—
115C4×C181—42no—
116C4×C1812−1+412yes—
117Dic181—41no—
118Dic181—41no—
119Dic181—41no—
120C2×C181—22no—
121C2×C181#53611yescharacteristic
122C2×C181C2×#13511yescharacteristic
123C2×C181#57511yescharacteristic
124D181—81no—
125C181#323211yescharacteristic, complemented, Sylow
126#3232—1811nomaximal, Sylow
127#529—1811no—
128C2×#125D18111yescharacteristic, complemented
129#499—1812no—
130#488—1812no—
131#484—1811no—
132#483—1811no—
133#481—1811no—
134#480—1811no—
135#479—1812no—
136#478—1812no—
137#431—1811no—
138#126—3621no—
139#125—21no—
140#125—21no—
141#125—1811no—
142#125C2×D18112yes—
143#125C2×D18112yes—
144#124—1811no—
145#123—3621no—
146C22⋊Q8—1811no—
147#121—1811no—
148C4×Q8—1811no—
149C4×D4—3621no—
150C4×D4—1811no—
151#117—3621no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×D18111yescharacteristic
155C2×C4⋊C4C2×D18111yescharacteristic
156C2×C42C2×D18111yescharacteristic
157#107—1812no—
158#107—1812no—
159#106—1812no—
160#106—1812no—
161C4⋊C8—1812no—
162#103—1812no—
163#103—1812no—
164#102—1812no—
165#102—1812no—
166C22⋊C8—1811no—
167C22⋊C8—1811no—
168C4×C8—1812no—
169C2×Q8—1811no—
170C2×D4—3621no—
171C2×D4—1811no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—3621no—
175C22×C4C22⋊D18111yescharacteristic
176C22×C4C22⋊D18111yescharacteristic
177C22×C4C22×D18111yescharacteristic
178C2×C8—1812no—
179C2×C8—1812no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—3621no—
187C4⋊C4—3621no—
188C4⋊C4—3621no—
189C4⋊C4—3621no—
190C4⋊C4—1811no—
191C4⋊C4—1811no—
192C4⋊C4C22×D18112yes—
193C4⋊C4C22×D18112yes—
194C22⋊C4—3621no—
195C22⋊C4—3621no—
196C22⋊C4—3621no—
197C22⋊C4—3621no—
198C22⋊C4—3621no—
199C42—3621no—
200C42C22⋊D18111yescharacteristic
201C42C22⋊D18111yescharacteristic
202C42C22×D18111yescharacteristic
203C42C22×D18111yescharacteristic
204C23—3621no—
205C23C2×C22⋊D18111yescharacteristic
206Q8—3622no—
207D4—3622no—
208D4—7242no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—3621no—
217C2×C4—3621no—
218C2×C4—3621no—
219C2×C4—7242no—
220C2×C4C2×C22⋊D18111yescharacteristic
221C2×C4C2×C22⋊D18111yescharacteristic
222C2×C4C2×C22⋊D18111yescharacteristic
223C2×C4C2×C22⋊D18111yescharacteristic
224C2×C4C2×C22⋊D18111yescharacteristic
225C2×C4C23×D18111yescharacteristic
226C8—3622no—
227C8—3622no—
228C22—21no—
229C22—21no—
230C22—7241no—
231C22—7241no—
232C22—7241no—
233C22#1915383612yes—
234C22C22×C22⋊D18111yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—7241no—
243C4—7241no—
244C4—7241no—
245C4#1915386412yes—
246C2—22no—
247C2—14481no—
248C2#2604185311yescharacteristic
249C2C2×#1915383611yescharacteristic
250C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#125×C181C2
socleC22×C181C22×D4
prime radical#19153790C22
hypercenterC2×#125D181
hypocenterC181#3232
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26041678C2semidirect
3. #26041636C2semidirect
4. #26041614C2semidirect
5. #26041609C2semidirect
8. #26041583C2semidirect
9. #26041576C2semidirect
10. #26041332C2semidirect
11. C2×#125×C181C2semidirect
125. C181#3232semidirect
128. C2×#125D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260418132
#260416782
#260416362
#260416142
#260416092
#260415972
#260415922
#260415832
#260415762
#260413322
C2×#125×C1812
#3232181